On time: punctual, not late.
If something happens ON time, it happens at the time which was planned:
'The 11.45 train left on time.' (= it left at 11.45)
'I'll meet you at 7.30' 'OK, but please be ON time.'
The opposite of ON time is 'late':
'Be on time. Don't be late.'
In time (for something / to do something) = soon enough
'I'm in a hurry. I want to be home IN time to see the game on television.'
= soon enough to see the game.
The opposite of IN time is 'too late'
at the same time / at the time / at the appoinyed time / at the usual time / at that time ...